A178678 Decimal expansion of the sum of alternating reciprocal square roots, omitting terms where n is a perfect square.

0, 8, 8, 2, 4, 8, 5, 3, 7, 1, 3, 8, 3, 1, 4, 9, 3, 9, 1, 6, 9, 9, 6, 6, 2, 0, 7, 2, 2, 2, 2, 2, 1, 0, 6, 8, 3, 1, 5, 7, 3, 7, 5, 8, 9, 2, 3, 0, 0, 0, 7, 8, 7, 3, 7, 4, 2, 1, 3, 3, 3, 6, 1, 4, 1, 1, 2, 0, 6, 3, 6, 8, 4, 7, 4, 6, 3, 4, 3, 5, 8, 2, 7, 8, 4, 5, 9, 3, 7, 0, 0, 7, 8, 0, 6, 9, 1, 3, 3, 1, 5, 8, 9, 6, 7

Offset

0,2

Comments

Provides a closed form for the Riemann zeta function of one half: Zeta(1/2) = (1 + sqrt(2))(R - log(2)).

The omitted sum of perfect squares equates to the natural logarithm of 2. Giving the alternating sum of all reciprocal square roots as log(2) - R.

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

Formula

R = Sum_{n>=2} (-1)^n/sqrt(n) for n that are not a perfect square.

R = 1/sqrt(2) - 1/sqrt(3) - 1/sqrt(5) + 1/sqrt(6) - 1/sqrt(7) + 1/sqrt(8) + ...

R = Sum_{n>=2} (-1)^(n+1)*(1-sqrt(n))/n.

Example

R=0.0882485371383149391699662072222210683157375892300078737421333614112...

Mathematica

RealDigits[(Sqrt[2] -1)*Zeta[1/2] +Log[2], 10, 100][[1]]

Prog

(PARI) default(realprecision, 100); (sqrt(2)-1)*zeta(1/2)+log(2) \\ G. C. Greubel, Jan 27 2019
(Magma) SetDefaultRealField(RealField(100)); L:=RiemannZeta();  (Sqrt(2)-1)*Evaluate(L, 1/2) +Log(2); // G. C. Greubel, Jan 27 2019
(SageMath) numerical_approx((sqrt(2)-1)*zeta(1/2)+log(2), digits=100) # G. C. Greubel, Jan 27 2019

Crossrefs

Cf. A059750, A002162, A002193.

Sequence in context: A224875 A242588 A105193 * A217459 A185280 A344074

Adjacent sequences: A178675 A178676 A178677 * A178679 A178680 A178681

Keyword

cons,nonn

Author

Matt Rieckman (mjr162006(AT)yahoo.com), Jun 03 2010

Extensions

Minor correction, simplified description, and additional comments Matt Rieckman (mjr162006(AT)yahoo.com), Jun 28 2010

Status

approved